$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Overview
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The unity operator basically changes a frame of reference
If $U^{\dagger}U=1$, where $U^{\dagger}$ is the Hermitian conjungate. For a constant operator, $U^{\dagger}$ is the complex conjungate of the matrix.

Notations

The unity operator is often denoted by a $U$. All that's required for an operator to be unity is $U^{\dagger}U=1$.

Concepts

Let $A$ be an arbitrary operator. Then $$\begin{align*} \langle A \rangle &= \langle \psi| A|\psi\rangle \\ &=\langle \psi|U^{\dagger}U A U^{\dagger} U|\psi\rangle \\ &=\langle \psi'| A' |\psi\rangle, \end{align*}$$ where $A'=UAU^{\dagger}$. Since $U^{\dagger}$ is the Hermitian conjungate of $U$, then $\langle \psi|U^{\dagger}=\langle \psi'|$ and $U|\psi'\rangle=|\psi'\rangle$, where $|\psi'\rangle$ is the new state in the new reference frame.
That is, if we transform a quantum state $|\psi\rangle$ by an unitary operator $U$ expectation values of an arbitary operator $A$ does not change as long as we transform $A$ into $A'=UAU^{\dagger}$.
For exaple, $$\begin{align*} U=\begin{bmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{bmatrix} \end{align*}$$ rotates a frame of reference by $\theta$.
The unity operator basically changes a frame of reference
If $U^{\dagger}U=1$, where $U^{\dagger}$ is the Hermitian conjungate. For a constant operator, $U^{\dagger}$ is the complex conjungate of the matrix.

Notations

The unity operator is often denoted by a $U$. All that's required for an operator to be unity is $U^{\dagger}U=1$.

Concepts

Let $A$ be an arbitrary operator. Then $$\begin{align*} \langle A \rangle &= \langle \psi| A|\psi\rangle \\ &=\langle \psi|U^{\dagger}U A U^{\dagger} U|\psi\rangle \\ &=\langle \psi'| A' |\psi\rangle, \end{align*}$$ where $A'=UAU^{\dagger}$. Since $U^{\dagger}$ is the Hermitian conjungate of $U$, then $\langle \psi|U^{\dagger}=\langle \psi'|$ and $U|\psi'\rangle=|\psi'\rangle$, where $|\psi'\rangle$ is the new state in the new reference frame.
That is, if we transform a quantum state $|\psi\rangle$ by an unitary operator $U$ expectation values of an arbitary operator $A$ does not change as long as we transform $A$ into $A'=UAU^{\dagger}$.
For exaple, $$\begin{align*} U=\begin{bmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{bmatrix} \end{align*}$$ rotates a frame of reference by $\theta$.
FullPage
Overview
Notations
Concepts
FullPage
Overview
Notations
Concepts
The unity operator basically changes a frame of reference
If $U^{\dagger}U=1$, where $U^{\dagger}$ is the Hermitian conjungate. For a constant operator, $U^{\dagger}$ is the complex conjungate of the matrix.

Notations

The unity operator is often denoted by a $U$. All that's required for an operator to be unity is $U^{\dagger}U=1$.

Concepts

Let $A$ be an arbitrary operator. Then $$\begin{align*} \langle A \rangle &= \langle \psi| A|\psi\rangle \\ &=\langle \psi|U^{\dagger}U A U^{\dagger} U|\psi\rangle \\ &=\langle \psi'| A' |\psi\rangle, \end{align*}$$ where $A'=UAU^{\dagger}$. Since $U^{\dagger}$ is the Hermitian conjungate of $U$, then $\langle \psi|U^{\dagger}=\langle \psi'|$ and $U|\psi'\rangle=|\psi'\rangle$, where $|\psi'\rangle$ is the new state in the new reference frame.
That is, if we transform a quantum state $|\psi\rangle$ by an unitary operator $U$ expectation values of an arbitary operator $A$ does not change as long as we transform $A$ into $A'=UAU^{\dagger}$.
For exaple, $$\begin{align*} U=\begin{bmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{bmatrix} \end{align*}$$ rotates a frame of reference by $\theta$.
The unity operator basically changes a frame of reference
If $U^{\dagger}U=1$, where $U^{\dagger}$ is the Hermitian conjungate. For a constant operator, $U^{\dagger}$ is the complex conjungate of the matrix.

Notations

The unity operator is often denoted by a $U$. All that's required for an operator to be unity is $U^{\dagger}U=1$.

Concepts

Let $A$ be an arbitrary operator. Then $$\begin{align*} \langle A \rangle &= \langle \psi| A|\psi\rangle \\ &=\langle \psi|U^{\dagger}U A U^{\dagger} U|\psi\rangle \\ &=\langle \psi'| A' |\psi\rangle, \end{align*}$$ where $A'=UAU^{\dagger}$. Since $U^{\dagger}$ is the Hermitian conjungate of $U$, then $\langle \psi|U^{\dagger}=\langle \psi'|$ and $U|\psi'\rangle=|\psi'\rangle$, where $|\psi'\rangle$ is the new state in the new reference frame.
That is, if we transform a quantum state $|\psi\rangle$ by an unitary operator $U$ expectation values of an arbitary operator $A$ does not change as long as we transform $A$ into $A'=UAU^{\dagger}$.
For exaple, $$\begin{align*} U=\begin{bmatrix}e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{bmatrix} \end{align*}$$ rotates a frame of reference by $\theta$.
FullPage
Overview
Notations
Concepts